I think there are 2 ways to get to Arbuckle Mountain. But I only did this one. I don’t recommend this route unless you are a 4x4 enthusiast or maybe by mountain bike.
From Willow Creek Road out of Heppner which turns into NF-53 then turn onto NF-170 (45.16615, -119.23569). This quickly turns into high clearance and eventually into 4wd road. However, it is only 3 miles to the top. I would not recommend this road if it is not dry. There are several turns required, but I did not document them as I took several wrong ones.